↑

Properties

Label 2-1152-72.11-c1-0-5
Degree $2$
Conductor $1152$
Sign $-0.426 - 0.904i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.524 − 1.65i)3-s + (1.86 + 3.22i)5-s + (−2.39 − 1.38i)7-s + (−2.44 + 1.73i)9-s + (−2.21 − 1.27i)11-s + (1.02 − 0.592i)13-s + (4.35 − 4.77i)15-s + 2.04i·17-s − 4.66·19-s + (−1.02 + 4.67i)21-s + (4.35 + 7.53i)23-s + (−4.44 + 7.70i)25-s + (4.14 + 3.13i)27-s + (2.70 − 4.67i)29-s + (−8.26 + 4.77i)31-s + ⋯
L(s)  = 1  + (−0.302 − 0.953i)3-s + (0.833 + 1.44i)5-s + (−0.905 − 0.522i)7-s + (−0.816 + 0.577i)9-s + (−0.668 − 0.385i)11-s + (0.284 − 0.164i)13-s + (1.12 − 1.23i)15-s + 0.497i·17-s − 1.07·19-s + (−0.223 + 1.02i)21-s + (0.907 + 1.57i)23-s + (−0.889 + 1.54i)25-s + (0.797 + 0.603i)27-s + (0.501 − 0.869i)29-s + (−1.48 + 0.856i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.426 - 0.904i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6092377666\)
\(L(\frac12)\) \(\approx\) \(0.6092377666\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.524 + 1.65i)T \)
good5 \( 1 + (-1.86 - 3.22i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.39 + 1.38i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.21 + 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.02 + 0.592i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.04iT - 17T^{2} \)
19 \( 1 + 4.66T + 19T^{2} \)
23 \( 1 + (-4.35 - 7.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.70 + 4.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.26 - 4.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.27iT - 37T^{2} \)
41 \( 1 + (7.62 - 4.40i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.97 - 5.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.39 - 4.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.13T + 53T^{2} \)
59 \( 1 + (3.79 - 2.18i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.02 + 0.592i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.60 - 9.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.91T + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 + (-8.26 - 4.77i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.92 + 1.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + (1.72 - 2.98i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23068871656144372399497389235, −9.396206563513343012924088130829, −8.209782563451453313435690365096, −7.33543552322616247886491828365, −6.65074250887358413261652941585, −6.14920808286872789953828412783, −5.32493376108696098999531097770, −3.50558581553854812124777998758, −2.81807804342207535351759463043, −1.67062009886565502586258220067, 0.25543286281754325383725383496, 2.10067442970872151458903701457, 3.32776766486789033644566244494, 4.67514504340716747383455482881, 5.05947996306976836464313941242, 5.98365010784371676503879563255, 6.72130351189661894113014316412, 8.380386148595386545448871165812, 8.908437977107286710550635356526, 9.455099648137861244773254098826

Graph of the $Z$-function along the critical line